1. Introduction

This course is about the topological structure of spaces. We start by discussing some examples. Here are some that you should remember from the Knots and Surfaces course: the cylinder, the sphere, the torus, the Möbius strip and the real projective plane.

The last picture actually shows a set called Boy’s surface, which crosses over itself. To understand how this relates to the real projective plane, consider (as an analogy) the following pictures:

The right hand picture shows the trefoil knot in ${\mathbb{R}}^{3}$, which does not intersect itself. The left hand picture is an attempt to represent the knot in two dimensions, but that is not possible without introducing self-intersections. Similarly, the real projective plane lives naturally in four dimensional space, where it has no self-intersections, and Boy’s surface is an imperfect three-dimensional representation.

In this course, we will consider many examples of spaces of dimension three or less, because that makes it easier to draw pictures. However, you should be familiar with the idea that any problem with $n$ variables can lead you to consider $n$-dimensional linear algebra, and thus sometimes to think about the geometry of lines, planes and so on in ${\mathbb{R}}^{n}$ and how they intersect. If we have nonlinear equations in $n$ variables, we may also need to consider nonlinear subspaces of ${\mathbb{R}}^{n}$ and their geometry and topology, and this can often have meaning in the real world even when $n>4$. In fact, in the recently developed field of Topological Data Analysis, it is common to have very large values of $n$. For example, there is active work on applications of TDA to neuroscience, in which $n$ is the number of neurons that one is modelling or monitoring. For large values of $n$ we cannot hope to draw meaningful pictures or rely on intuition. Instead, we need methods that convert problems from topology into more tractable questions in algebra. This is the main goal of Algebraic Topology.

Example 1.1.

One family of examples that we can use to illustrate this goal consists of the letters of the alphabet. We will regard these as subsets of the plane ${\mathbb{R}}^{2}$, drawn using infinitely thin lines as follows:

Many of these (such as $C$, $L$ and $W$) can be straightened out to just give a line segment. With a bit more bending and stretching we can make the $E$, $F$, $J$ and $Y$ look the same as the $T$.

We can redraw the table as follows:

On the left, we have drawn all the letters again in red, grouping together letters that have similar properties. On the right, we have drawn a straightened-out form for each letter. It seems that all the letters in the first group are topologically equivalent, as are all the letters in the second group. However, the letters in the first group do not seem to be equivalent to those in the second group. How can we make this precise? One approach uses the following definition:

Definition 1.2.

A cut point of $X$ is a point $x\in X$ such that $X\setminus\{x\}$ is disconnected. We define $d(X)$ to be the number of points that are not cut points.

For the first three groups of letters, almost all the points are cut points; the only exceptions are the tips of the various branches. Thus, for the first group of letters we have $d=2$, for the second group we have $d=3$, and for the third group we have $d=4$. However, for the letters $D$, $O$ and $B$ there are no cut points, so $d=\infty$. For the remaining letters the situation is more complicated, but there are still infinitely many points where you can cut without disconnecting the space, so again $d=\infty$.

What can we take away from this discussion?

• (a)

We have used some ideas about connected and disconnected spaces that seem very reasonable, but we still have not made precise definitions or proved any theorems, which will be essential if we want to make sure that everything will work in higher dimensions.

• (b)

We have implicitly assumed that topologically equivalent spaces have the same value of $d(X)$. This is true if we use the most obvious version of topological equivalence, which is called homeomorphism, but we need to give an actual proof of that. We will also spend a lot of time discussing a different notion called homotopy equivalence. It will turn out that homotopy equivalent spaces need not have the same value of $d(X)$, which emphasises why we need to be careful with definitions and proofs.

• (c)

We have defined a number $d(X)$, such that $d(X)=d(Y)$ whenever $X$ and $Y$ are homeomorphic. In other words, $d(X)$ is a numerical homeomorphism invariant of $X$. The letters $B$ and $C$ have different values of $d$, so they are not homeomorphic. (This is visually obvious, but now we have a method of proof that we can hope to apply in cases that are not visually obvious.)

• (d)

However, this logic does not work backwards. The letters $H$ and $X$ both have $d=4$, but we cannot conclude that these letters are homeomorphic. In fact, we can prove that they are not homeomorphic: if we remove the central point from $X$ then the remaining space breaks into $4$ connected pieces, but there is no way to break $H$ into more than $3$ pieces by removing a single point.

• (e)

Moreover, this technique becomes very ineffective if we consider more complicated spaces. For example, we can remove any finite set of points from the sphere and it will still remain connected, and the torus has the same property, so this approach has no chance of detecting the difference between the sphere and the torus. We will need other techniques that need much more work to set up.

Now consider again the letters $B$, $O$ and $C$. Although we can distinguish between them by methods similar to those described above, this is in some sense missing the most obvious point: $B$ has two holes, $O$ has one hole, and $C$ has none. Unfortunately, it will take a great deal of work to give a proper mathematical formulation of this point, and we will not achieve that until a long way into the course. However, we will explain one interesting aspect now. Consider the following pictures:

The right hand picture shows a space $X$ in three dimensions, consisting of three lines of longitude joining the north and south poles of the sphere $S^{2}$. The left hand picture is the same space, flattened out into the plane. Looking at the right hand picture, it seems natural to say that there are three holes, arranged symmetrically around the $z$-axis. Looking at the left hand picture, it seems natural to say that there are only two holes. Which is correct? The key point here is that we should not think of the holes as just giving a number; instead, there is an abelian group of holes, called $H_{1}(X)$, the first homology group of $X$. There really are three symmetrically arranged holes, which we can call $a$, $b$ and $c$, but they satisfy $a+b+c=0$. We can therefore use the relation $c=-a-b$ to express every element of $H_{1}(X)$ in the form $na+mb$ for some $(n,m)\in{\mathbb{Z}}^{2}$, so the group $H_{1}(X)$ is isomorphic to ${\mathbb{Z}}^{2}$. This is the sense in which there are really only two basic holes. The notion of addition and subtraction used here is not obvious, but it will emerge naturally when we give the formal definitions.

This should hopefully motivate the idea that our invariants should not just be numbers, but should instead be algebraic structures such as groups or rings. We will eventually define abelian group denoted by $H_{n}(X)$ for all $n\geq 0$, which are again called homology groups. For example, the torus $T$ has $H_{0}(T)\simeq H_{2}(T)\simeq\mathbb{Z}$ and $H_{1}(T)\simeq{\mathbb{Z}}^{2}$ and $H_{n}(T)=0$ for $n>2$. We will also write $H_{*}(X)$ for the sequence of all homology groups of $X$, so $H_{*}(T)=({\mathbb{Z}},{\mathbb{Z}}^{2},{\mathbb{Z}},0,0,\dotsc)$ for example. We can use these groups to prove many interesting topological facts that are completely inaccessible by other methods.

Our approach to homology groups will involve simplices and simplicial complexes.

Definition 1.3.

The standard $n$-simplex is the space

 $\Delta_{n}=\{(x_{0},\dotsc,x_{n})\in{\mathbb{R}}^{n+1}\;|\;x_{i}\geq 0\text{ % for all }i\text{ and }\sum_{i}x_{i}=1\}.$

The vertices of $\Delta_{n}$ are just the standard basis vectors $e_{0},\dotsc,e_{n}$, so $e_{0}=(1,0,\dotsc,0)$ and $e_{1}=(0,1,0,\dotsc,0)$ and $e_{n}=(0,\dotsc,0,1)$ and so on.

The standard $0$-simplex $\Delta_{0}$ is just a single point, and $\Delta_{1}$ is a line segment, and $\Delta_{2}$ is a triangle, and $\Delta_{3}$ is a tetrahedron. We can draw them as follows:

A key technique will be to study spaces by dividing them up into simplices. Consider the following pictures:

The middle picture shows the undivided sphere $S^{2}$, which is what we really want to study. The second picture shows $S^{2}$ divided into curved triangles in an octahedral pattern, and the fourth picture shows $S^{2}$ divided into curved triangles in an icosahedral pattern. The first picture is a genuine octahedron with flat faces, and the last picture is a genuine icosahedron with flat faces. These are examples of simplicial complexes. The sphere is homeomorphic to both the octahedron and the icosahedron. We will eventually prove a theorem that will allow us to use the combinatorial structure of either of these complexes to compute the homology of $S^{2}$ (although in this example, other methods of computing homology are easier).

Example 1.4.

Here are some further examples of higher-dimensional spaces that we will consider later.

• (a)

The most basic example is ${\mathbb{R}}^{n}$. Although it might seem obvious, it is already difficult to prove that ${\mathbb{R}}^{n}$ and ${\mathbb{R}}^{m}$ are not homeomorphic when $n\neq m$, but we will achieve that by the end of the course.

• (b)

Inside ${\mathbb{R}}^{n+1}$ we have the unit ball and the unit sphere:

 $\displaystyle B^{n+1}$ $\displaystyle=\{x\in{\mathbb{R}}^{n+1}\;|\;\|x\|\leq 1\}$ $\displaystyle S^{n}$ $\displaystyle=\{x\in{\mathbb{R}}^{n+1}\;|\;\|x\|=1\}.$

(The last picture is supposed to depict a hollow shell; the space inside does not count as part of $S^{2}$.)

• (c)

We have already seen the simplex $\Delta_{n}$. This is actually homeomorphic to the ball $B^{n}$, or to the cube $[0,1]^{n}$. Later we will see a nice general theorem that makes it easy to prove that something is homeomorphic to $[0,1]^{n}$. Alternatively, we can draw a picture for $n=2$: there is a homeomorphism sending the numbered points to the numbered points and the dotted edges to the dotted edges.

We will also be interested in skeleta of simplices:

 $\operatorname{skel}^{k}(\Delta_{n})=\{x\in\Delta_{n}\;|\;\text{ at least }n-k% \text{ coordinates are zero }\}$

For example, the $1$-skeleton of $\Delta_{n}$ consists of all the vertices and edges of $\Delta_{n}$, but nothing else.

• (d)

The $n$-dimensional torus is $S^{1}\times\dotsb\times S^{1}=(S^{1})^{n}$. The case $0$-dimensional torus is just a point, the $1$-dimensional torus is a circle, and the $2$-dimensional torus is what we normally just call the torus.

• (e)

Let $M_{n}({\mathbb{R}})$ be the space of $n\times n$ matrices over the real numbers. This can be identified with the space ${\mathbb{R}}^{n^{2}}$ which we have considered already. Inside $M_{n}({\mathbb{R}})$, we can consider the subspace $GL_{n}({\mathbb{R}})$ of invertible matrices, and the subspace $O_{n}$ of orthogonal matrices (satisfying $A^{T}A=I$), and the subspace $SO_{n}$ of special orthogonal matrices (satisfying $\det(A)=1$ as well as $A^{T}A=I$). All of these spaces have interesting topology. For example, $GL_{2}({\mathbb{R}})$ is homeomorphic to ${\mathbb{R}}^{3}\times S^{1}\times\{1,-1\}$ (as shown in Example 4.11), but the answers are more complicated for $n>2$. There are also similar examples involving complex matrices.

• (e)

The $n$-dimensional real projective space ${\mathbb{R}}P^{n}$ is obtained from the sphere $S^{n}$ by identifying $x$ with $-x$ for all $x$. This is a kind of quotient construction, for which we will need to recall various ideas about equivalence relations and study how they interact with topology. (This is also required for a rigorous treatment of the kind of gluing constructions that you will have seen in the Knots and Surfaces course.) It works out that ${\mathbb{R}}P^{1}$ is homeomorphic to $S^{1}$ and ${\mathbb{R}}P^{3}$ is homeomorphic to the matrix group $SO_{3}$. There is also a different way to describe ${\mathbb{R}}P^{n}$ as a space of matrices: it turns out that ${\mathbb{R}}P^{n}$ is homeomorphic to the space

 $P_{n}=\{A\in M_{n+1}({\mathbb{R}})\;|\;A^{2}=A^{T}=A,\;\text{trace}(A)=1\}.$
• (f)

We can also consider the complex projective space ${\mathbb{C}}P^{n}$, for this, we recall that $S^{2n+1}$ is the unit sphere in ${\mathbb{R}}^{2n+2}$, but ${\mathbb{R}}^{2n+2}$ can be identified with ${\mathbb{C}}^{n+1}$, so the points of $S^{2n+1}$ can be regarded as complex vectors. This lets us form a quotient space in which $x$ is identified with $zx$ whenever $z$ is a complex number with $|z|=1$. This quotient space is ${\mathbb{C}}P^{n}$. It turns out that ${\mathbb{C}}P^{1}$ is just $S^{2}$, but ${\mathbb{C}}P^{2}$ is already quite interesting.

In order to understand all these examples, we need a general theory of topology. The framework of metric spaces is adequate for most, but not all purposes. We will therefore spend some time on the more general framework of topological spaces. This is a large and important topic in its own right, but we will try to cover the minimum that we need without too many distractions.